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June 12, 2007

Spectral Triples and Graph Field Theory

Posted by Urs Schreiber

Had the pleasure of talking with Yan Soibelman over dinner. He mainly educated me about his old work with Kontsevich concerning 2-dimensional CFTs and their strings-collapse-to-points limit, which is described in section 2 of

Yan Soibelman is trying to paint in fuller detail a beautiful picture, in which Connes’ spectral triples are the algebraic data encoding a quantum mechanical system (called a “graph field theory” here, since it computes amplitudes of Feynman graphs) which is obtained from a 2-dimensional conformal field theory (describing a stringy object instead of a point particle) by a limiting procedure in which

\;\; a) the circumference of the string shrinks to zero

\;\; b) the volume of the target space that the string propagates in grows without bounds (“large volume limit”).

At its core, this question is an extremely old hat for physicists, in a way, since it is really the very ∼30\sim 30 year old motivation for looking at string theory in the first place: generalize Feynman diagrams from graphs to tubular diagrams.

But attempts to make many things here precise are scarce. And attempts doing so using Connes’ spectral geometry (what Connes calls “noncommutative geometry”) are even more scarce — but they shouldn’t be.

It seems to me that what Yan Soibelman is carrying on is the study of stringy spectral geometry (or 2-NCG / 2-spectral geometry, as I think it should ultimately amount to), which people started looking at in old work like

In a way, this is all about understanding functors
QFT:nCob→nVect
QFT : n\mathrm{Cob} \to n\mathrm{Vect}
using only algebraic (as opposed to geometric) tools and data – and then interpreting that data geometrically.

For instance, one of the main questions Yan Soibelman said he is currently thinking about is how to conceive the notion of Ricci curvature, and in particular of bounded from below Ricci curvature, in terms of Connes’ spectral geometry. There is supposedly some deep relation between spectral triples obtained from the point particle limit of 2-dimensional CFTs and the Ricci curvature of the spectral geometry which they encode.

The basic idea is simple and rather obvious:

Our 2-dimensional conformal field theory sends conformal cobordisms to linear maps between vector spaces of states assigned to their boundaries.

Differentiating this assignment on the conformal cylinder – regarded as a parallelogram with diagonal τ∈ℂ\tau \in \mathbb{C} in the complex plane – with respect to the real part of τ\tau, yields the “generator of infinitesimal time translation”, which is usually denoted
L0+L¯0
L_0 + \bar L_0
and called “the sum of the left and right part of the zero mode of the Virasoro algebra”.

This operator L0+L¯0L_0 + \bar L_0 plays essentially the role that the Hamiltonian does in the quantum mechanics of the point particle. Only that here we have the quantum mechanics of the string. In fact, to a large degree L0+L¯0L_0 + \bar L_0 may be regarded as something like a Laplace operator on loop space.

At least if we think of the CFT here as describing a string which literally propagates in an ordinary space. In this situation we would say that the CFT arises from a σ\sigma-model (of maps from Riemann surfaces into target space) and that the string background is in a geomtric phase.

The analogous situation of this “geometric phase” would be a functor 1CobRiem→Hilb1\mathrm{Cob}_{\mathrm{Riem}} \to \mathrm{Hilb}, which sends 1-dimensional Riemannian cobordisms of length tt to the operator
e−tL,
e^{-t L}
\,,
where LL is the Laplace operator of some Riemannian space. It need not. We get a 1-dimensional QFT, a functor 1CobRiem→Hilb1\mathrm{Cob}_{\mathrm{Riem}} \to \mathrm{Hilb}, for any positive hermitean operator LL. But if LL is a Laplace-like operator on some truly geometric space, then we’d say the background of the particle described by our functor is in a geometric phase.

Now, one important phenomenon is that 2-dimensional CFTs may not exactly come from such “geometric phases”, but that they approximate these closely.

A good 1-dimensional analogy here is actually Connes’ latest spectral triple model for the world we inhabit: this is a spectral space which is close to an ordinary geometric space. In fact, it is such that a single superparticle propagating through this spetral geometry looks like a bunch of different species of particles (namely all elementary particles which we see in particle accelerators, plus the Higgs particle which we expect to see with the new LHC), all of them propagating on an ordinary Minkowski space.

Anyway, the question Soibelman (based on Kontsevich & Soibelman) asks is: how do we understand the limit in which our 2-dimensional QFT2Cobconf→2Hilb
2\mathrm{Cob}_{\mathrm{conf}} \to 2\mathrm{Hilb}
degenerates to a 1-dimensional QFT1CobRiem→Hilb
1\mathrm{Cob}_{\mathrm{Riem}} \to \mathrm{Hilb}
such that the latter comes from a spectral triple with a commutative algebra – hence being pretty close to an ordinary geometry.

The limit to be taken is quite clear heuristically: we want to shrink the string to a point. This means, given any Riemann surface swept out by the string, we want to decompose it into lots of trinions (three-holed spheres) and lots of cylinders stretching between these, and then let the length of these cylinders tend to infinity.

For this to make sense, we need at the same time let the “target space valume tend to infinity”. Technically, this means that we let the “mass gap”, namely the first nonvanishing eigenvalue of L0+L¯0L_0 + \bar L_0 tend to zero.

What one arrives at this way is in fact a little more than just a functor
1CobRiem→Hilb.
1\mathrm{Cob}_{\mathrm{Riem}} \to \mathrm{Hilb}
\,.
Such a functor is defined only on worldlines which are disjoint unions of intervals or circles – since these are the only 1-dimensional manifolds there are!

But the string actually had interactions, where it split into two strings or two merged into one: these were the trinions. In the limit, these trinions do flow to something finite which remains: a certain interaction on a trivialent vertex of a graph.

This way, Kontsevich and Soibelman actually obtain what they call a “graph field theory”, a functor on something like
1CobRiembranch
1\mathrm{Cob}^{\mathrm{branch}}_{\mathrm{Riem}}
whose morphisms are graphs which become 1-dimensional Riemannian cobordisms once all vertices are removed.

(The fact that such funny business is not required for 2Cob2\mathrm{Cob} (unless we consider limits like here) has traditionally been regarded as an indication for the naturalness of the idea of passing from 1-particles to 2-particles.)

What is remarkable about this is the following:

as long as we are just looking at functors
1CobRiem→Hilb,
1\mathrm{Cob}_{\mathrm{Riem}} \to \mathrm{Hilb}
\,,
we find that these are given by just two thirds of a spectral triple: a Hilbert space of states and a Hamiltonian (the Laplace operator).

Where is the last third: the algebra AA represented on that Hilbert space?

Of course Kontsevich and Soibelman now find this realized in the additional ingredient which they have: the (3-valent) vertices!

Apparently for this to make sense we have to assume that there is not just a representation of AA by bounded operators on our Hilbert space, but actually a dense embedding of our algebra (regarded as a vector space) into the Hilbert space.

(Of course this is the case in the standard motivating example where A=C(X)A = C(X) and H=L2(X)H = L^2(X)).

Then we can take the map
H⊗H→H
H \otimes H \to H
which our functor associated to the 3-valent vertex and read it as a map
A⊗A→A,
A \otimes A \to A
\,,
and that’s then the product on our algebra.